Development and Validation of a Quasi-one-dimensional Particle-in-cell Code for Magnetic Nozzle SimulationFrans H. Ebersohn1, JP Sheehan1, Benjamin W. Longmier1, John V. Shebalin2

1Aerospace Engineering, University of Michigan; 2Astromaterials Research Office, NASA Johnson Space Center

5th Annual MIPSE Graduate Student Symposium

The formulation and validation of a novel quasi-one-dimensional particle-in-cell code for the simulation of magnetic nozzles is presented.

Strong guiding magnetic fields known as magnetic nozzles are key components in the design of electrodeless plasma thrusters. The operating regime of many magnetic nozzle devices (ex. Cubesat Ambipolar Thruster) lie near the edge of the continuum regime, making numerical simulation challenging.[1]

We developed a novel quasi-one-dimensional electrostatic particle-in-cell method which focuses on studying energy exchange and thermalization in the plasma. Electron-Electron Two Steam Instability

The code is validated with standard one-dimensional test cases. A quasi-one-dimensional method for magnetic nozzle simulation is developed and its implementation shows promising results for magnetic mirror test cases. Further evaluation of Q1D algorithms and implementation will be done in the future.

This research is funded by a NASA Space Technology Research Fellowship. Thank you to my fellow lab mates at PEPL for their helpful discussions and support.

[1] Ebersohn, Frans H., et al. "Magnetic nozzle plasma plume: review of crucial physical phenomena." AIAA paper 4274 (2012): 2012. [2] Boris, J. P. "Relativistic plasma simulation-optimization of a hybrid code." Proc. Fourth Conf. Num. Sim. Plasmas, Naval Res. Lab, Wash. DC. 1970.[3] Birdsall, Charles K., and A. Bruce Langdon. Plasma physics via computer simulation. CRC Press, 2004.[4] Denavit “Nonrandom Initializations of particle codes,” Comments Plasma Phys. Control. Fusion 6, 209-223, September 1981.[5] Cartwright, K. L., J. P. Verboncoeur, and C. K. Birdsall. "Loading and injection of Maxwellian distributions in particle simulations." Journal of Computational Physics 162.2 (2000): 483-513.[6] Schwager, Lou Ann, and Charles K. Birdsall. "Collector and source sheaths of a finite ion temperature plasma." Physics of Fluids B: Plasma Physics (1989-1993) 2.5 (1990): 1057-1068.

Governing equations:𝑑𝑑𝒙𝒙𝑑𝑑𝑑𝑑 = 𝒗𝒗

𝑚𝑚𝑑𝑑𝒗𝒗𝑑𝑑𝑑𝑑

= 𝑞𝑞(𝑬𝑬 + 𝒗𝒗 × 𝑩𝑩)Particle Movers:• Standard Boris Algorithm for Axisymmetric Coordinates[2],[3]• Modified Semi-Implicit Q1D Boris Algorithm

𝒗𝒗𝑛𝑛+12−𝒗𝒗𝑛𝑛−

12

Δ𝑡𝑡= 𝑞𝑞

𝑚𝑚𝑬𝑬 + 𝒗𝒗

𝑛𝑛+12+𝒗𝒗𝑛𝑛−12

2× 𝑩𝑩 + 𝒂𝒂𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐

𝑛𝑛+12 +𝒂𝒂𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑛𝑛−12

2

𝒂𝒂𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐+12 =

𝑣𝑣𝜃𝜃𝑐𝑐+12

2

𝑟𝑟𝐿𝐿�̂�𝑟 −

𝑣𝑣𝑟𝑟𝑐𝑐+12𝑣𝑣𝜃𝜃

𝑐𝑐+12

𝑟𝑟𝐿𝐿�̂�𝜃

𝑟𝑟𝐿𝐿 =𝑚𝑚𝑣𝑣𝜃𝜃𝑞𝑞𝐵𝐵𝑧𝑧�̂�𝑧

Magnetized particles are assumed to be displaced from the axis by their Larmorradii to incorporate axial magnetic forces and 2D effects.

𝐵𝐵𝑟𝑟 = −𝑟𝑟𝐿𝐿2𝜕𝜕𝐵𝐵𝑧𝑧𝜕𝜕𝑧𝑧

The cross sectional area varies by assuming that particles approximately follow magnetic field lines.

𝐴𝐴 = 𝐵𝐵𝑧𝑧,𝑖𝑖𝑛𝑛𝐵𝐵𝑧𝑧

𝐴𝐴𝑖𝑖𝑐𝑐

Landau Damping

Velocity phase space and energy time history comparisons with results of Birdsall.[3]

Electrostatic energy time history comparison with Denavit.[4]

Comparison of magnetic nozzle to De Laval nozzle.[1]

Illustration of the model domain overlaid on a firing of the Cubesat Ambipolar Thruster (CAT).

Source and Collector Plasma Sheath

Magnetic Mirror

Comparison between our simulations and those of Schwager[6] for a source and collector sheath.

Particle motion in a magnetic bottle. Initial and final velocity distribution function in magnetic mirror.

The centerline axis of the magnetic nozzle is modeled and three velocity dimensions are resolved.

Electrostatic Particle-In-Cell algorithm.

Initial and final velocity space for magnetic mirror simulation with analytical loss cone (blue).

Left: Bmax = 1.75 TRight: Bmax= 4.0 T

Abstract

Methodology

Introduction

Quasi-one-dimensional Algorithms

Validation Simulations

Conclusions

Acknowledgements

Development and Validation of a Quasi-one-dimensional Particle-in-cell Code for Magnetic Nozzle Simulation�Frans H. Ebersohn1, JP Sheehan1, Benjamin W. Longmier1, John V. Shebalin2�1Aerospace Engineering, University of Michigan; 2Astromaterials Research Office, NASA Johnson Space Center